This is something I found out late last year with my black Family Edition Wii (RVL-101) but it can also be done on the Wii mini (RVL-201). I thought I'd share it here in case someone had one of these models and didn't know.
The modding process for the Wii mini is a bit more involved but is possible thanks to BlueBomb. The Family Edition Wii can be modded easily using BannerBomb. After that you just need to install Nintendont and you should be able to use Xbox 360, Xbox One, PS3, PS4, Switch Pro, and several other kinds of controllers to play GameCube games.
This post on [email protected] titled Playing GameCube classics, but in 4K on a Mac with PlayStation controllers, just as Nintendo intended. was what reminded me of this. If you are into the Wii [email protected] and [email protected] seem to be the other two Wii communties on Lemmy